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Probability of finding a particle between a and b at time t 

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Calculating expectation values for < x > and < x^2 > shortcuts 

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odd function with x > goes to 0 even function > double and change bounds 


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E_n for infinite square well 

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[image]
f(x) is the wavefunction (generic) 


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Probability of states in well 

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Observables are represented by 

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Eigenvalue eqn for Q operator 

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Q Ψ = q Ψ
Ψ = eigenfunction q = eigenvalue 


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Integration by parts formula 

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Normalizeable eigenfunctions of a hermitian operator 

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are orthogonal (int f*g = 0) and their eigenvalues are real 


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Expectation value of Q_op 

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Generalized uncertainty principle 

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The system is initially in state (a+), and you want probability of (b) 

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Allowed energy for a hydrogen atom 

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Label spacial wavefunctions for hydrogen 

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What quantum numbers are "allowed" for a given n? 

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l = 0,1,2,...,n 1 m_l = l,l+1,...,0,...,l1,l m_s = ±1/2 


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Hamiltonian in perturbation theory 

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Superscripts in perturbation theory 

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Order. Ex: H^0 first order 


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First order energy correction 

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[image]
This is an integral H' * Psi^2
Can also be X (eigenstate) 


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First order wavefunction correction 

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Two non interacting particles wavefunction 

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Atom placed in uniform external B field 

